Download 11-2 Simplifying Radical Expressions - lindsey-math

11-1 Simplifying Radical Expressions
**Radical Expression: An expression that contains a
square root.
**Square Root: Think: “What times itself is that
number” (ex: √25 = 5 because 5 x 5 = 25)
**When you have a square root number, and it is
NOT a perfect square, think “what times what is that
number” and write it under the √. THEN, see if one
of those numbers is a perfect square!
EXAMPLE: √12
Step 1: 12 is not a perfect square, but 4 x 3 = 12,
so write 4 x 3 inside the √ symbol.
√4 x 3
Step 2: the square root of 4 is 2. Write the 2 on
the OUTSIDE of the √ , and keep the 3 inside!
2√3
EXAMPLE: √3 × √15
Step 1: First multiply √3 × √15 = √45
Step 2: Put 9 × 5 under the √
Step 3: take the square root of 9, put it on the
outside, and keep the 5 on the inside.
3√5
**For radical expressions where the variable’s
exponent is EVEN, and the resulting simplified
exponent is ODD, then the answer must have
ABSOLUTE VALUE signs!
√x² = |x|
√x³ = x√x
√x^4 = x²
√x^5 = x²√x
√x^6 = |x³|
√x^7 = x³√x
**When there is a √ on the denominator, multiply the
numerator and denominator by the denominator.
Then simplify!
EXAMPLE: √10
√3
Step 1: multiply √3 to both √10 and √3
Step 2: simplify √10 × √3 = √30 = √30
√3
√3
√9
3
**Conjugates: Something like 3 + √2 and 3 - √2
(similar to (x – 5)(x + 5) = x² - 25)
EXAMPLE:
2
6 - √3
Step 1: multiply the top and bottom by 6 + √3
Step 2: Use FOIL to multiply it
2
× 6 + √3 = 2(6 + √3)
6 - √3 6 + √3 36 + 6√3 - 6√3 - √9
Step 3: Simplify
12 + 2√3 = 12 + 2√3 = 12 + 2√3
36 - √9
36 - 3
33
11-2 Operations with Radical Expressions
**If the radicands (numbers under the √) are the
same, then you can add these expressions (similar to
adding like terms.)
EXAMPLE: 4√3 + 6√3 - 5√3
Step 1: since they are all √3, add the 4, 6, and -5
together (and keep the √3) and get 5√3.
EXAMPLE: 12√5 + 3√7 + 6√7 - 8√5
Step 1: Add “like terms” 12√5 - 8√5…..and 3√7 +
6√7….. and get 4√5 + 9√7.
EXAMPLE: 2√20 + 3√45 + √180
Step 1: simplify each radical expression to see if
they have the same radicand (the # under the √) so you
can add them (to simplify….see 11-1 notes)
2√20 = 4√5
3√45 = 9√5
√180 = 6√5
Step 2: Add them to get 19√5
**To multiply radical expressions, it is similar to
multiplying by using the distributive property (or
“FOIL” for some problems)
EXAMPLE: √6(√3 + 5√2)
Step 1: multiply the √6 by √3 and 5√2 and get
√18 + 5√12
Step 2: Simplify each radical (see 11-1 notes)
3√2 + 10√3
EXAMPLE: (3 + √5)(3 - √5)
Step 1: multiply the first 3 by everything that’s in
the 2nd parenthesis and get
9 - 3√5
Step 2: then multiply the √5 by everything that’s
in the 2nd parenthesis and get
3√5 - √25
Step 3: simplify all those numbers
9 - 3√5 + 3√5 - √25 = 9 - √25
Step 4: simplify that number to
9–5=4
11-3 Radical Equations
**When you have a radical in an equation, treat the
radical like an “x-term”…and get it by itself on one
side of the equal sign. THEN… SQUARE both sides
____
EXAMPLE: √x + 1 + 7 = 10
Step 1: subtract 7 from both sides to get the
radical by itself.
_____
√x + 1 = 3
Step 2: square both sides (when you square a
radical, it just ELIMINATES the √ sign)
_____
(√x + 1 )²= (3)² after you square it, you get:
x+1=9
Step 3: finish solving for x
x=8
Step 4: Put x = 8 back into the original problem to
make sure it is TRUE!!! _____
√8 + 1 + 7 = 10
√9 + 7 = 10
3 + 7 = 10 YES!!!!
____
EXAMPLE: √x + 2 = x – 4
Step 1: the radical is already by itself, so just
SQUARE both sides
____
(√x + 2)² = (x – 4)²
Step 2: When you square the √ side, just get rid of
the √ symbol. When you square (x – 4), use FOIL
x + 2 = (x – 4)(x – 4)
x + 2 = x² - 4x – 4x + 16
x + 2 = x² - 8x + 16
Step 3: The x² makes it a Quadratic Formula,
where it HAS TO EQUAL 0!!!! Make it equal zero!
x + 2 = x² - 8x + 16
-x -2
-x
-2
0 = x² - 9x + 14
Step 4: Solve (however you solve it!) (8-steps, tic
tac no-toe, graph, quadratic equation….)
(x – 7)(x – 2) = 0
Step 5: Solve for BOTH values of x
x–7=0
x–2=0
+7 +7
+ 2 +2
x=7
x =2
Step 6: You have to plug EACH one of your xvalues into the ORIGINAL equation b/c only ONE
will be your answer
____
√7 + 2 = 7 – 4
√2 + 2 = 2 – 4
√9 = 3
√4 = -2
3 = 3 YES!!
2 ≠ -2 NO!!!
7 is the ONLY solution!!!
11-4 The Pythagorean Theorem
11- 5 The Distance Formula
The distance formula: d = √(x – x)² + (y – y)²
To use the distance formula:
Step 1: Substitute your x’s and y’s.
Step 2: Subtract them (separately).
Step 3: Square them.
Step 4: Add them together
Step 5: Take the square root
EXAMPLE: Find the distance between the points at
(2, 3) and (-4, 6)
Step 1: Substitute your x’s and y’s into the
equation.
√(2 – (-4))² + (3 – 6)²
Step 2: Subtract your parenthesis
√(6)² + (-3)²
Step 3: Square each one
√36 + 9
Step 4: Add
√45
Step 5: Take the square root
6.71